Important Questions For CBSE Class 12 Maths Chapter 3
Important Questions For CBSE Class 12 Maths Chapter 3
Important Questions For CBSE Class 12 Maths Chapter 3
x 3 4 5 4
1. If , find x and y.
y 4 x y 3 9
i 0 0 i
2. If A and B , find AB .
0 i i 0
2i – j if i j
where aij .
i 2 j 3 if i j
5 2 3 6
5. If A and B 0 1 find 3 A 2B.
0 9
2 3 1 0
6. If A and B find A B ´.
7 5 2 6
2
7. If A = [1 0 4] and B 5 find AB .
6
4 x 2
8. If A is symmetric matrix, then find x.
2x 3 x 1
0 2 3
2 0 4 is skew symmetrix matrix.
9. For what value of x the matrix
3 4 x 5
2 3
10. If A P Q where P is symmetric and Q is skew-symmetric
1 0
matrix, then find the matrix Q.
1
a ib c id
11. Find the value of c id a ib
2x 5 3
12. If 0, find x .
5x 2 9
k 2
13. For what value of k, the matrix has no inverse.
3 4
2 3 5
15. Find the cofactor of a12 in 6 0 4 .
1 5 7
1 3 2
16. Find the minor of a23 in 4 5 6 .
3 5 2
1 2
17. Find the value of P, such that the matrix is singular.
4 P
18. Find the value of x such that the points (0, 2), (1, x) and (3, 1) are
collinear.
19. Area of a triangle with vertices (k, 0), (1, 1) and (0, 3) is 5 unit. Find the
value (s) of k.
20. If A is a square matrix of order 3 and |A| = – 2, find the value of |–3A|.
22. What is the number of all possible matrices of order 2 × 3 with each entry
0, 1 or 2.
23. Find the area of the triangle with vertices (0, 0), (6, 0) and (4, 3).
2x 4 6 3
24. If , find x .
1 x 2 1
2
x y y z z x
y , write the value of det A.
25. If A z x
1 1 1
a11 a12
26. If A such that |A| = – 15, find a11 C21 + a12C22 where Cij is
a21 a22
cofactors of aij in A = [aij].
5 3
28. If A find adj A
6 8
29. Given a square matrix A of order 3 × 3 such that |A| = 12 find the value
of |A adj A|.
2 1
32. If A find A 1 1 .
3 4
3
33. If A 1 2 3 and B 4 find |AB|.
0
x y 2x z –1 5
34. Find x, y, z and w if .
2x y 3x w 0 13
3
1 2 3 3 0 1
36. Find A and B if 2A + 3B = and A 2B .
2 0 1 1 6 2
1
2 and B 2 1 4 , verify that (AB)´ = B´A´.
37. If A
3
3 3 1
38. Express the matrix 2 2 1 P Q where P is a symmetric and Q
4 5 2
is a skew-symmetric matrix.
2 1 5 2 2 5
40. Let A , B , C , find a matrix D such that
3 4 7 4 3 8
CD – AB = O.
1 3 2 1
41. Find the value of x such that 1 x 1 2 5 1 2 0
15 3 2 x
5 3 2 –1
43. If A show that A – 12A – I = 0. Hence find A .
12 7
4
2 3 2
44. If A find f(A) where f(x) = x – 5x – 2.
4 7
4 3
45. If A , find x and y such that A2 – xA + yI = 0.
2 5
1 2 3 7 8 9
46. Find the matrix X so that X 2 4 6 .
4 5 6
2 3 1 2 –1 –1 –1
47. If A and B then show that (AB) = B A .
1 4 1 3
48. Test the consistency of the following system of equations by matrix
method :
3x – y = 5; 6x – 2y = 3
49. Using elementary row transformations, find the inverse of the matrix
6 3
A , if possible.
2 1
3 1
50. By using elementary column transformation, find the inverse of A .
5 2
cos sin
51. If A and A + A´ = I, then find the general value of .
sin cos
Using properties of determinants, prove the following : Q 52 to Q 59.
a b c 2a 2a
3
52. 2b b c a 2b a b c
2c 2c c a b
x 2 x 3 x 2a
53. x 3 x 4 x 2b 0 if a, b, c are in A.P .
x 4 x 5 x 2c
5
2 2 2 2
b c a a
2 2 2 2 2 2 2
55. b c a b 4a b c .
2 2 2 2
c c a b
b c c a a b a b c
56. q r r p p q 2 p q r .
y z z x x y x y z
2 2
a bc ac c
2 2 2 2 2
57. a ab b ac 4a b c .
2 2
ab b bc c
x a b c
2
58. a x b c x x a b c .
a b x c
59. Show that :
x y z
2 2 2
x y z y z z x x y yz zx xy .
yz zx xy
60. (i) If the points (a, b) (a´, b´) and (a – a´, b – b´) are collinear. Show
that ab´ = a´b.
2 5 4 3
(ii) If A and B verity that AB A B .
2 1 2 5
0 1
0 1 2
61. Given A and B 1 0 . Find the product AB and
2 2 0
1 1
also find (AB)–1.
62. Solve the following equation for x.
a x a x a x
a x a x a x 0.
a x a x a x
6
0 tan 2
63. If A and I is the identity matrix of order 2, show
tan 0
2
that,
cos sin
I A I A
sin cos
65. Obtain the inverse of the following matrix using elementary row operations
0 1 2
A 1 2 3 .
3 1 1
1 1 2 2 0 1
0 2 3 9 2 3 to solve the system of equations
66. Use product
3 2 4 6 1 2
x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2.
2 3 3 1 1 1 3 1 2
10, 10, 13.
x y z x y z x y z
1 2 3
68. where A 2
F ind A–1, 3 2 , hence solve the system of linear
3 3 –4
equations :
x + 2y – 3z = – 4
2x + 3y + 2z = 2
3x – 3y – 4z = 11
7
69. The sum of three numbers is 2. If we subtract the second number from
twice the first number, we get 3. By adding double the second number
and the third number we get 0. Represent it algebraically and find the
numbers using matrix method.
3 1 1
A 15 6 5 and verify that A–1 A = I3.
5 2 5
1 1 2 1 2 0
A 0 3 and B 0
–1 ,
–1
71. If the matrix 2 3 then
3 2 4 1 0 2
compute (AB)–1.
72. Using matrix method, solve the following system of linear equations :
2x – y = 4, 2y + z = 5, z + 2x = 7.
0 1 1
A 2 3I
73. Find A 1
if A 1 0 1 . Also show that A 1 .
2
1 1 0
1 2 2
0 by using elementary
74. Find the inverse of the matrix A 1 3
0 2 1
column transformations.
2 3 2
75. Let A and f(x) = x – 4x + 7. Show that f (A) = 0. Use this result
1 2
to find A5.
cos sin 0
76. If A sin cos 0 , verify that A . (adj A) = (adj A) . A = |A| I3.
0 0 1
8
2 1 1
77. For the matrix A 1 2 1 , verify that A3 – 6A2 + 9A – 4I = 0, hence
1 1 2
find A–1.
3 2 1 1 2 1
7 5 . X .
2 1 0 4
1 a2 b 2 2ab 2b
3
2ab 1 a2 b 2 2a 1 a 2 b 2 .
2b 2a 1 a2 b 2
y z 2 xy zx
2 3
80. xy x z yz 2xyz x y z .
xz yz x y 2
a ab a b c
81. 2a 3a 2b 4a 3b 2c a 3 .
3a 6a 3b 10a 6b 3c
x x 2 1 x 3
82. If x, y, z are different and y y 2 1 y 3 0. Show that xyz = – 1.
z z2 1 z 3
83. If x, y, z are the 10th, 13th and 15th terms of a G.P. find the value of
log x 10 1
log y 13 1 .
log z 15 1
9
84. Using the properties of determinants, show that :
1 a 1 1
1 1 1
1 1 b 1 abc 1 abc bc ca ab
a b c
1 1 1 c
85. Using properties of determinants prove that
bc b 2 bc c 2 bc
3
a 2 ac ac c 2 ac ab bc ca
a 2 ab b 2 ab ab
3 2 1
86. If A 4 1 2 , find A–1 and hence solve the system of equations
7 3 3
3x + 4y + 7z = 14, 2x – y + 3z = 4, x + 2y – 3z = 0.
ANSWERS
0 1
1. x = 2, y = 7 2. 1 0
3. 11. 4. 4
9 6 3 5
5. 0 29 . 6. 3 1 .
7. AB = [26]. 8. x = 5
0 1
9. x = – 5 10. 1 .
0
3
13. k 14. |A| = 1.
2
15. 46 16. –4
10
5
17. P = – 8 18. x .
3
10
19. k . 20. 54.
3
25. 0 26. 0
8 3
27. 9 28. 6 5 .
3 3 2 5 2
4 5 2 .
35.
5 6 7
11 9 9 5 2 1
7 7 7 7
7 7
36. A , B
1 18 4 4
12 5
7 7 7 7 7 7
191 110
40. D . 41. x = – 2 or – 14
77 44
7 3
43. A 1 . 44. f(A) = 0
12 5
1 2
45. x = 9, y = 14 46. x .
2 0
11
48. Inconsistent 49. Inverse does not exist.
2 1
50. A –1 . 51. 2n , n z
5 3 3
1 2 –1 1 2 2
61. AB , AB .
2 2 6 2 1
11 1
62 0, 3a 64. x , y .
24 24
1 1 1
–1 2 2 2
65. A 4 3 1 . 66. x = 0, y = 5, z = 3
5
3 1
2 2 2
6 17 13
1 1 1 1
67. x , y , z 68. A 1
14 5 8
2 3 5 67
15 9 1
2 0 1
69. x = 1, y = – 2, z = 2 70. A 1
5 1 0
0 1 3
16 12 1
71. AB 1 1 21 11 7 . 72. x = 3, y = 2, z = 1.
19
10 2 3
1 1 1 3 2 6
1
73. A 1
1 1 1 . 74. A 1
1 1 2
2
1 1 1 2 2 5
12
118 93 3 1 1
A5 1
1 .
1
75. . 77. A 1 3
31 118 4
1 1 3
16 3
78. X . 83. 0
24 5
86. x = 1, y = 1, z = 1.
13